On semilinear problems with nonlinearities depending only on derivatives. (English) Zbl 0852.34018

The authors consider semilinear boundary value problems \[ u''(t)+ \lambda_1 u(t)+ g(t, u'(t))= f(t),\quad t\in I,\tag{1} \]
\[ (Bu)(t)= 0,\quad t\in \partial I,\tag{2} \] where \(I= [0, \pi]\), \(B\) denotes either the Dirichlet or the Neumann or the periodic boundary conditions, respectively, and \(\lambda_1\) is the first eigenvalue of the corresponding linear problem \(u''(t)+ \lambda u(t)= 0\), \(t\in I\), \((Bu)(t)= 0\), \(t\in \partial I\). The nonlinear function \(g\) is supposed to be bounded and, in some cases, satisfies additional differentiability assumptions and asymptotic conditions. The authors emphasize the dependence of \(g\) on the derivative of the solution \(u'(t)\) in order to show the qualitative difference of this case and the Landesman-Lazer-type problem in which the nonlinearity \(g\) depends only on the solution \(u(t)\). The authors establish the solvability of the problem (1), (2).


34B15 Nonlinear boundary value problems for ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
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